cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A080499 Duplicate of A072513.

Original entry on oeis.org

1, 1, 2, 6, 4, 60, 6, 168, 48, 360, 10, 47520, 12, 1092, 1680, 20160, 16, 440640, 18
Offset: 1

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Author

Keywords

A080497 a(n) = (n-p_1)(n-p_2)...(n-p_k) where p_k is the k-th prime and is also the largest prime < n.

Original entry on oeis.org

1, 1, 1, 2, 6, 12, 40, 90, 336, 840, 1728, 3150, 10560, 24948, 99840, 270270, 604800, 1201200, 4386816, 11277630, 49029120, 143896500, 348364800, 746876130, 2937876480, 8117240040, 18923520000, 39628338750, 76859228160, 140548508100
Offset: 1

Views

Author

Amarnath Murthy, Mar 19 2003

Keywords

Examples

			a(6) = (6-2)(6-3)(6-5) = 12. a(7) = (7-2)(7-3)(7-5) = 40.

Crossrefs

Cf. A080498, A072513, A080500.

Programs

  • Mathematica
    a[n_] := Product[n - Prime[k], {k, 1, PrimePi[n - 1]}]; Array[a, 30] (* Amiram Eldar, Dec 01 2018 *)
  • PARI
    a(n) = my(mk = primepi(n-1)); prod(k=1, mk, n-prime(k)); \\ Michel Marcus, Dec 01 2018

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 06 2003

A080498 a(n) = (n-c_1)(n-c_2)...(n-c_k) where c_k is the k-th composite number and is also the largest composite number < n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 8, 15, 48, 210, 1152, 3780, 19200, 62370, 322560, 2162160, 17418240, 81081000, 567705600, 2481078600, 16907304960, 146659312800, 1504935936000, 8799558768000, 76435881984000, 819678899239200
Offset: 1

Views

Author

Amarnath Murthy, Mar 19 2003

Keywords

Examples

			a(6) = (6-4) = 2. a(10) = (10-4)(10-6)(10-8)(10-9) = 48.

Links

Crossrefs

Cf. A080497, A072513, A080500.

Programs

  • Mathematica
    Module[{nn=30,cmps},cmps=Select[Range[nn],CompositeQ];Table[Times@@(n-Select[ cmps,#Harvey P. Dale, Aug 13 2024 *)

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 06 2003

A080500 a(n) = (n-1)(n-4)(n-9)...(n-k^2) where k^2 < n <= (k+1)^2.

Original entry on oeis.org

1, 1, 2, 3, 4, 10, 18, 28, 40, 54, 140, 264, 432, 650, 924, 1260, 1664, 4284, 8100, 13376, 20400, 29484, 40964, 55200, 72576, 93500, 236808, 443232, 728000, 1108380, 1603800, 2235968, 3028992, 4009500, 5206760, 6652800, 8382528, 20867704
Offset: 1

Views

Author

Amarnath Murthy, Mar 19 2003

Keywords

Comments

The idea of A080497 to A080500 when applied to Euler's phi function i.e. on phi-torial function defined in A001783 yields A001783 itself for obvious reasons. Is there any other such example?

Examples

			a(6) = (6-1)(6-4)= 10.

Crossrefs

Cf. A080497, A080498, A072513.

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 06 2003

A258324 Least common multiple of all n - d, where d < n and d is a divisor of n.

Original entry on oeis.org

1, 1, 2, 6, 4, 60, 6, 84, 24, 360, 10, 3960, 12, 1092, 420, 840, 16, 12240, 18, 13680, 1260, 4620, 22, 1275120, 120, 7800, 936, 19656, 28, 1096200, 30, 52080, 5280, 17952, 7140, 5654880, 36, 25308, 8892, 2489760, 40, 1343160, 42, 397320, 27720
Offset: 1

Views

Author

Ivan Neretin, May 26 2015

Keywords

Comments

a(n) is a divisor of A072513(n).
a(n) = n-1 if and only if n is prime. - Robert Israel, May 26 2015

Examples

			a(9) = lcm(9-1, 9-3) = lcm(8, 6) = 24.

Links

Crossrefs

Cf. A072513 (product instead of LCM).
Cf. A027751.

Programs

  • Haskell
    a258324 n = foldl lcm 1 $ map (n -) $ a027751_row n
-- Reinhard Zumkeller, May 27 2015
  • Maple
    f:= n -> ilcm(seq(n-d, d = numtheory:-divisors(n) minus {n})):
map(f,[$ 1 .. 100]); # Robert Israel, May 26 2015
  • Mathematica
    Table[If[n == 1, 1, LCM @@ (n - Most[Divisors[n]])], {n, 50}]
  • PARI
    a(n)=lcm(apply(d->if(dCharles R Greathouse IV, May 26 2015

Formula

a(n) = lcm(n-d_1, n-d_2, ..., n-d_k) where d_i are the aliquot divisors of n.

A072512 Product of all n - d, where 1 < d < n and d is a divisor of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 12, 1, 24, 6, 40, 1, 4320, 1, 84, 120, 1344, 1, 25920, 1, 43200, 252, 220, 1, 31933440, 20, 312, 432, 183456, 1, 136080000, 1, 322560, 660, 544, 840, 12563527680, 1, 684, 936, 919296000, 1, 1155772800, 1, 1219680, 1814400, 1012, 1
Offset: 1

Views

Author

Amarnath Murthy, Jul 28 2002

Keywords

Comments

For prime p, a(p) = 1.
If n is not a prime or the square of a prime then n divides a(n).

Examples

			For n = 16 the nontrivial divisors d are 2,4 and 8, so a(16) = (16-2)*(16-4)*(16-8) = 14*12*8 = 1344.

Links

Crossrefs

Cf. A072513.

Programs

  • Maple
    f:= proc(n) local d; mul(n-d, d = numtheory:-divisors(n) minus {1,n}) end proc:
map(f, [$1..50]); # Robert Israel, Dec 30 2024
  • PARI
    a(n) = my(d=divisors(n)); prod(j=2, matsize(d)[2]-1, n-d[j]);

Extensions

Edited and extended by Klaus Brockhaus, Jul 31 2002

A380600 Irregular table T(n, k), n > 0, k = 1..A000005(n) read by rows: the n-th row lists the numbers of the form n * (d-1) / d with d a positive divisor of n.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 2, 3, 0, 4, 0, 3, 4, 5, 0, 6, 0, 4, 6, 7, 0, 6, 8, 0, 5, 8, 9, 0, 10, 0, 6, 8, 9, 10, 11, 0, 12, 0, 7, 12, 13, 0, 10, 12, 14, 0, 8, 12, 14, 15, 0, 16, 0, 9, 12, 15, 16, 17, 0, 18, 0, 10, 15, 16, 18, 19, 0, 14, 18, 20, 0, 11, 20, 21, 0, 22
Offset: 1

Views

Author

Rémy Sigrist, Feb 02 2025

Keywords

Examples

			Table T(n, k) begins:
  n   n-th row
  --  ------------------
   1  0
   2  0, 1
   3  0, 2
   4  0, 2, 3
   5  0, 4
   6  0, 3, 4, 5
   7  0, 6
   8  0, 4, 6, 7
   9  0, 6, 8
  10  0, 5, 8, 9
  11  0, 10
  12  0, 6, 8, 9, 10, 11
  13  0, 12
  14  0, 7, 12, 13

Links

Crossrefs

Cf. A000005, A027750, A046666, A060681, A072513, A094471 (row sums), A258324.

Programs

  • Mathematica
    Table[Map[n*(# - 1)/# &, Divisors[n]], {n, 23}] // Flatten (* Michael De Vlieger, Feb 03 2025 *)
  • PARI
    row(n) = apply (d -> n*(d-1)/d, divisors(n))

Formula

T(n, k) = n * (A027750(n, k) - 1) / A027750(n, k).
Sum_{k = 1..A000005(n)} T(n, k) = A094471(n).
Product_{k = 2..A000005(n)} T(n, k) = A072513(n).
LCM{k = 2..A000005(n)} T(n, k) = A258324(n).
T(n, 1) = 0.
T(n, 2) = A060681(n) for any n > 1. - Michel Marcus, Feb 03 2025
T(n, A000005(n)-1) = A046666(n) for any n > 1.
T(n, A000005(n)) = n-1.
Showing 1-7 of 7 results.

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